Balancing a three-phase power transmission system for an electric arc furnace



m 3 1968 J. WATTERSON BALANCING A THREE-PHASE POWER TRANSMISSION SYSTEMFOR AN ELECTRIC ARC FURNACE 21, 1964 2 Sheets-Sheet l Filed Dec.

INVENTOR. H F/V/QV J. M77Z/PJDA BY [QM/v 4 J 3 1968 v H. J. WATTERSONBALANCING A THREE-PHASE POWER TRANSMISSION SYSTEM FOR AN ELECTRIC ARCFURNACE Filed Dec'. 2 Sheets-Sheet United States Patent Henry J.Watterson, Rocky River, Ohio, assignor t0 Watteredge Co., Rocky River,Ohio, a corporation of Ohio Filed Dec. 21, 1964, Ser. No. 419,902 27Claims. (Cl. 13-12) This invention relates to electrical transmissionsystems and more particularly to a transmission system adapted forsupplying poly-phase power at high current to an electrical load, suchas an electric arc furnace.

The electrical transmission system supplying an electric arc furnace issubject to a number of physical and electrical requirements. The poweris fed from a main power source by means of a high voltage transmissionline to a transformer which transforms the power to one of relativelylow voltage and high current; for example, currents of the order oftwenty thousand to seventy-five thousand emperes per phase or greaterand voltages of the order of one hundred to five hundred volts. Theoutput of the transformer is coupled through transmission lines toelectrodes in the arc furnace. These electrodes must be capable of beingmoved vertically to control the arc length, to permit the metal to beintroduced to the furnace, and to permit the furnace to be relined. Incertain furnaces, it is necessary that the furnace be capable of beingrotated from a vertical to an inclined position to pour the melt. Also,electric arc furnaces are frequently provided with covers which aremounted for pivotal movement in a horizontal plane to charge the furnaceand to permit the furnace lining to be replaced. Thus, it is importantthat the transmission system include flexible cables which permitrelative movement between the electrodes and movement of the electrodesrelative to the furnace when the furnace is being charged or when thelining is being replaced and installations which require the furnace tobe rotated. One of the problems in transmission of power to electric arcfurnaces is that of obtaining and maintaining efficient operation whilesatisfying the above mentioned physical requirements.

From the electrical standpoint, numerous difficulties are encountered inproviding a transmission system for electrical arc furnaces. Smalldifferences in the impedances of the phase groups cause differences inthe order of megawatts in the power delivered to the several arcs. Whenattempts are made to correct for the differences by adjusting the arclength, one of the arcs may produce sufficient heat to ruin the furnacelining while a second arc may produce insufficient heat to melt thesurrounding scrap and the third electrode may short out to the melt. Itis well known that imbalances between phases of a three-phase system canbe caused by differences in the self-inductance and resistance within aphase group and the mutual inductance between phase groups ofconductors. For example, differences in the geometric mean radius, orGMR, of the conductors of one phase of a three-phase system relative tothe conductors of the other phases produces imbalances in the selfinductance of the one phase groups of conductors relative to the otherphase groups. It is also well known that imbalances of mutual inductanceexist between the phase gruops of conductors of a poly-phase system ifdifferences in the effective spacing exist between the phase groups ofconductors, which distances are known as the geometric mean distances,or GMDs. This imbalance is minimized if the phase groups of conductors,as viewed in cross-section, are placed with their centers at the cornersof an equilateral triangle.

This idealized equilateral spacing is not always practical V ice orpossible for a number of reasons. For example, the electrode bus barsconnected to the furnace electrodes connect to the electrodes, in aco-planar relationship. In electrical transmission of high currents,however, a coplanar relationship of conductors produces imbalances dueto the differences in geometric mean distances between phase groups andthe resulting mutual inductances between the several lines. Because ofthe previously mentioned physical requirements, it is difficult toemploy a transmission line for an electric arc furnace in which thisequilateral spacing of lines is maintained. Ernst lifatent 2,908,736tends to disclose an arc furnace transmission system provided with busbars and cables having different GMRs by having differentcross-sectional areas between bus bars and different cross-sectionalareas between the cables in an effort to provide equal mutualinductances between the phase groups. This solution, however, introducesan imbalance between the resistances of the phase groups and does notresult in equal power to each electrode. Dillon et al. Patent 3,078,325tends to disclose cables in which the central group of cables is longerthan the two outer phase groups so that a triangular relationship ofphase groups of cables tends to exist over a portion of the cablelength. This partial triangulation does not produce a sufficientimpedance balance between the several phase groups to produce equalpower at the electrodes. These known systems exhibit certain otherdisadvantages. For example, they require the stocking of cables ofdifferent length or different cross-sectional area, which iseconomically impractical. Although the electric arc furnace transmissionsystems are relatively short, for example, fifty feet or less and theimbalances of resistance and reactance are quite small, i.e., of theorder of microohms per foot of conductor, the resulting imbalance inpower to the respective arcs is quite large because the current in eachphase is of the order of twenty thousand to seventy-five thousandamperes per phase.

Accordingly, it is an object of this invention to provide an improvedthree-phase power transmission system with interchangeable cables eachcontaining conductors of equal length and cross-section, which system isadapted to transmit relatively high currents with equality of currentvalues in the different phases.

Another object of this invention is to provide a balanced three-phasepower transmission system employing the same sizes and lengths of cablesin each phase and which is particularly adapted to transmit equal powerto the separate arcs of an electric arc furnace.

A further object of this invention is to provide a substantiallybalanced power transmission system for an electric arc furnace whichemploys cables which are interchangeable between all the phase groups.

Yet a further object of this invention is to provide a poly-phase powertransmission system for an electric arc furnace which deliverssubstantially equal power to each arc of the furnace.

Still another object of this invention is to provide a powertransmission system for an electric arc furnace which employs conductorsof the same length and crosssectional area for each phase and whichtransfers the maximum useful power to all the electrodes.

Yet another object of this invention is to provide a power transmissionsystem for an electric arc furnace which increases the rate of furnacemelt.

A still further object of this invention is to provide a powertransmission system for an electric arc furnace which increases theefficiency of the furnace by equalizing the melting time in each of thearcs.

Another object of this invention is to provide an electric arc furnacetransmission system which increases the refractory life.

A still further object of this invention is to provide a three-phasepower transmission system for an electric arc furnace which results in areduction of electric consumption.

Yet a further object of this invention is to provide an electric arcfurnace power transmission system which reduces the power required perton of melt.

It is still a further object of this invention to provide a powertransmission system for an electric arc furnace in which interchangeablecables having the same lengths and cross sectional areas of conductorsare employed in each phase group and in which the power to the arcs isequalized by varying the geometric mean radii of at least two phasegroups.

Yet another object of this invention is to provide balanced power in thearcs of a three-phase arc furnace by using as a transmission system,phase groups of conductors of the same length and cross-sectional areaand by varying the geometric mean radii and the geometric mean distancesof the conductors of the system.

Still another object of this invention is to provide a poly-phaseelectric arc furnace power transmission system in which equal sizes andlengths of conductors are employed and in which the power is balanced byvarying the geometric mean distances between phase groups of conductors.

I have noticed that if three electrodes of a three-phase electric arcfurnace are supplied from co-planar threephase lines having conductorphase groups A, B, C in which the phase sequence is A, B and C, thefirst or leading phase, phase A, will have a relatively cold arc; thesecond, or intermediate phase, B, will have the hottest arc; and thetrailing, or remaining phase, C, will have an arc of intermediate heat,relative to the arcs of phases A and B. In accordance with the preceptsof a preferred form of my invention, maximum power transfer to the arc,maximum furnace refractory life and maximum efficiency of the system canbe achieved by equalizing the power transmitted to the arcs of thefurnace. In my preferred form, I employ equal conductors, i.e., of thesame length and cross-sectional area. In a preferred form of myinvention, I adjust the geometric mean radius of the conductors bymodifying the spacing between equal conductors Within a single phase. Inthis form, I also employ non-co-planar groups of conductors and adjustgeometric mean distance between phase groups. I prefer to make these twoadjustments with respect to an initial assumed co-planar relationshipand employ either an equal or an unequal number of equal conductors ineach phase. By so doing, my transmission system will transmitsubstantially equal power to each of the arcs.

I am aware of the fact that with an A, B, C phase sequence, co-planarconductors of a poly-phase system connecting an electrode for each phasesupply the A phase, or leading phase, electrode with the least power,the B phase, or intermediate phase, electrode with the most power, andthe trailing, or C phase, electrode with an intermediate amount ofpower. This power imbalance exists even though the electrodes are underthe control of a current and voltage responsive system which maintainsthe arc lengths within certain limits. Space limitations within thefurnace transmission system area may prevent or render costly orimpractical, the placing of the flexible or rigid conductors of thetransmission system in the form of an equilateral triangle. It ispossible, however, to achieve, or advantageously approach, an equalpower transfer to each of the arcs by employing interchangeableconductors of the same length and cross-sectional area in each phasegroup; a spacing between the conductors of the transmission system inwhich certain ratios of geometric mean distances and geometric meanradii are varied with respect to the power delivered to the respectiveelectrodes by these same conductors positioned in a co-planartransmission system. An understanding of these ratios and theirrelationships to the voltage drops in the transmission system can bestbe illustrated by a mathematical analysis of the transmission system.The voltage drops per centimeter for the A, B, and C phase groups ofconductors of a three-phase system, expressed in terms of conductorspacing, are:

. 1 1 +2 7-I wln Tug-P212100) 111 DAC Where:

In the secondary transmission system of an electric arc furnace, i.e.between the transformer secondary and the furnace electrodes, there aretwo sections. The first section is the flexible cable section and inthis invention employs cables of equal length, L (expressed incentimeters). The second section is the rigid bus section preferablyemploying tubes or bars of equal length, L (expressed in centimeters).Assuming that equal currents are flowing in the conductors of eachgroup, then the following conditions prevail:

I I I is the absolue value of current in each group of conductors; and

V V and V and V V V are the first and second section voltage drops ofthe A, B and C conductor groups, respectively.

If the relationships set forth in (4) and (5) exist, then the realvalues of the voltage drops on opposite sides of the equality signs mustbe equal and the imaginary values on opposite sides of the equalitysigns must be equal. By inserting the vectoral expressions for equalthree-phase currents (with the appropriate sign) and employing theidentity:

( 1 A LOG +Loo Loo The relationships of the real values expressed in (4)becomes:

An 'Ao L2 The relationship of the imaginary values in (4) becomes:

DBDBC) L AB 1 The relationship of the real values in (5) becomes:

The relationship of the imaginary values in (5) becomes:

'AB 'Ao L D' 2 DCDBC L1 DACZ DAC2

Equations {7 through 10 provide the basis for a number of conclusionsregarding the eifect of the spacing between phase groups of conductors.For example, because the ratio of the distances between phase groupsappears in the real Equations 7 and 9, it is apparent that changes inthe ratio of these distances affect the resulting resistance of one ormore phase groups. Further, the voltage drop may be increased bydecreasing the distance expressed in the denominators of the ratios inthat equation. Further, certain of these values may be held constantwhile certain other values are modified such that the resultant changein the voltage drop will be either directly or inversely proportional tothat quantity which is modified. For example, if the resistancesexpressed in Equation 7 are maintained constant, the lengths L and Lmaintained constant, then the equality may be obtained by either varyingthe distance D or the distance D Assuming that the bus is of constantspacing, then the balance must be achieved by adjusting either thedistance D or D in the flexible cables. Assuming that the A phasevoltage drop is less than the B phase voltage drop, equality can beachieved by either increasing D or decreasing D Normally, because ofphysical limitations and because of the positioning of the transformersecondary terminals, the distance AC is not varied but the distance ABmay be conveniently varied. Accordingly, in this particular example, theimpedance or the voltage drop in the A phase, may be increased bychanging the distance D in an increasing direction. This can beconveniently achieved by letting the A phase group conductors remainstationary and moving the B phase conductors farther away from the Aphase group. The variation of the one variable may be sufiicient toproduce the balance. Alternatively, it is possible to vary the spacingbetween the bus groups D' and D' to provide the desired balance. It isunderstood, however, that the range within which these dimensions can bevaried is restricted by the position of the electrodes and by the heightof the building which restricts the height to which the intermediategroup of buses may be elevated.

The term co-planar as employed herein shall be construed to refer to therelationship, in any perpendicular cross-section, of the centers ofgravity of the phase groups of conductors. For example, I havepreviously employed transmission systems in which phase groups of equalor unequal numbers of interchangeable conductors have been positionedwith the centers of gravity, or geometric centers, within an inch or twoof a common horizontal plane when the conductors are viewed inperpendicular section at any point along their length. This co-planararrangement produced a less satisfactory distribution of power than thenon-co-planar systems of the subject invention. In the subjectinvention, I employ a non-coplanar relationship preferably positioningthe outer phase groups of conductors with their geometric centers in afirst horizontal plane and position the intermediate, or middle, phasegroup with its geometric center in a plane remote from the first planesuch that the geometric centers are located at the corners of atriangle. For example, I position the outer phase groups with theirgeometric centers four feet apart and in a first horizontal plane andposition the middle phase group with its center of gravity in a secondplane three feet from the first plane. The terms leading and lagging asemployed herein apply to the phase sequence of voltage applied to theouter phase groups of conductors. For example, if the phase groups aredesignated A, B, and C from left to right, as viewed in section andlooking toward the electrodes, and the voltage wave applied to the Aphase electrodes precedes the voltage applied to the C phase (regardlessof the position of the B phase voltage wave) then the A phase is theleading phase and the C phase is the trailing or lagging phase.Conversely, if the voltage wave applied to the C phase group precedesthe voltage wave applied to the A phase group then the C phase group isthe leading phase and the A phase is the trailing phase.

Other objects and features and advantages of the present invention willbe more fully apparent to those skilled in the art from the followingdetailed description taken in connection with the accompanying drawingin which:

FIG. 1 is a diagrammatic, fragmentary, schematic perspective of anelectric arc furnace with an illustrative power transmission system;

FIG. 2 is a plan view of a preferred bus arrangement;

FIGS. 3a and 3b are views in section, taken along the lines 33 of FIG. 2and showing two bus arrangements;

FIGS. 4a and 4b are views in section, taken along the lines 4-4 of FIG.1 and showing two embodiments of flexible cable configuration accordingto this invention.

Referring now to the drawing, FIG. 1 is a diagrammatic schematicrepresentation of a three-phase power transmission system for supplyingpower to an electric arc furnace, which system includes a transformer 10which receives power from a suitable transmission line, not shown, andsupplies this power to an electric arc furnace 12. The transformer 10includes a set of transformer terminals 13, 14, and 15, one for eachphase, to which is connected a set of bus bars 17, 18, and 19,respectively. For the purposes of this explanation, it will be assumedthat terminals 13, 14, and 15 are for the A phase, or leading phase theB phase or the intermediate phase, and the C phase or trailing phase,respectively. In other words, it is assumed that the phase sequence ofthe transmission system supplying the transformer 10 has a phasesequence A, B, C and that terminals 13, 14, and 15 are the A, B C phasesecondary output terminals, respectively. Any other phase sequence canbe employed so long as the relationship between leading and laggingphases and the respective disposition of conductors is retained. The

power supplied to the buses 17, 18, and 19 is to be supplied to threeelectrodes 29, 21, and 22, respectively of the furnace 12 through twosections of conductors. The first section of conductors are individualphase groups of flexible cables, the A phase group of which is indicatedby a series of short dashed lines 23, a second or B phase group offlexible cables represented by a series of alternate long and shortdashed lines 24, and the third, or C phase group of flexible cablesrepresented by a series of long dashes 25. These flexible cable groups23, 24, and are connected to a second section of conductors which arepreferably phase groups of rigid buses 27, 28, and 29 for the A, B, andC phase, respectively. It is the relative size, length and dispositionof the conductors 23, 24 and 25 of the flexible type and the rigid busconductors of the bus groups 27, 28 and 29 which constitute the subjectmatter of this invention. 1 have discovered that if the phase groups ofconductors of the flexible type, designated 23, 24 and 25, are co-planaras determined at spaced sectional views, along their length and if therigid buses 27, 28 and 29 are also co-planar, then the power deliveredto the electrodes 20, 21 and 22 will be unbalanced and the leading phaseelectrode, which in this example is the A phase, electrode 20, will havethe coldest arc, the intermediate or B phase electrode 21 will have thehottest arc and the trailing or C phase electrode 22 will receive anamount of power intermediate that of electrodes 20, 21. Thisrelationship of power distribution prevails even though the electrodesare under control of an electrode position control system which respondsto both current and voltage supplied to the electrodes of the type wellknown in the art. For example, this electrode position control systemmay be of the type produced by the Westinghouse Electric Corporation anddisclosed on page 17 of the Iron and Steel Daily News, Sept. 23, 1964, apublication of the Association of Iron and Steel Engineers. Theseelectrode position and control systems are capable of vertically varyingthe electrode positions individually and can adjust these electrodepositions within relatively narrow limits in an attempt to equalize thepower distribution to the several electrodes. The electrode positioncontrol system, however, cannot vary over a sufiicient range tocompensate for substantial differences in the self inductance and themutual inductance of a transmission system.

FIG. 2 is a plan View, partly broken away, of a preferred bar or bustube arrangement and the connections of the flexible cables thereto andthe connections of the buses with the furnace electrodes 20, 21, and 22.The bus bars are connected in A, B, and C phase groups and in thisexample, include three conductors in each phase group. The B phase groupis straight while the A and C phase groups have parallel sections 30,divergent sections 31 and convergent sections 32. Phase group 27 of theA phase includes buses 33, 34, and 35 which are positioned with theiraxes at the corners of a triangle as shown in FIGS. 3a and 3b, whichfigures show alternative arrangements of these bus bars. These buses areconnected between a hanger plate 36 mounted on a suitable verticallymovable frame, not shown, which can be elevated and lowered to raise andlower electrode 20. Similarly, the B phase buses 28 include three,preferably cylindrical, bus bars or tubes 39, 40 and 41 spaced withtheir axes at the corners of a triangle and connected between a hangerplate 43 mounted on a vertically movable platform, also not shown, underthe control of the previously described electrode position controlequipment and a terminal 44 mounted on the electrode 21. The C phase busgroup 29, is similar to the buses 27 for the A phase and these buses areconnected between a hanger plate 46 and an electrode terminal connector47 on electrode 22 and include bus tubes 48, 49, and 50. As shown inFIG. 2, the flexible conductors indicated as 23, 24, and 25 for the A,B, and C phases, respectively, include three conductors of equalcross-sectional area and preferably these conductors are of equallength. The A phase group 23 includes conductors 52, 53, and 54. Theseconductors are of the same cross-sectional area and length as the Bphase group of conductors 24, indicated as including flexible conductors55, 56, and 57 which are connected to the hanger plate 43. Similarly,the C phase group of flexible conductors 25 is connected to the hangerplate 46 and these conductors include the conductors 58, 59, and 60.These flexible conductors 58, 59, and 60 are the same cross-sectionalarea and length as the conductors 55, 56, and 57 and as the conductors52, 53, and 54 and all the flexible conductors may be forced air cooledbut are preferably water cooled. I have discovered that by controllingthe spacing between the conductors within a phase group, known as thegeometric mean radius, and by controlling the distances between thecenters of gravity of the phase groups, known as the geometric meandistances, a substantially equal power distribution to the arcs ofelectrodes 20, 21, and 22 can be achieved.

FIG. 3a is a View in section, taken along the lines 3--3 of FIG. 2 andshowing in section, triangular positional relationships between the bustubes of each phase group and showing the non-co-planar triangularrelationship between phase groups. Preferably, the buses 27, 28, and 29and conductors 23, 24, and 25 are each enclosed in a large flexibletube, not shown, through which cooling water is forced. The facilitiesfor supplying water to the conductors are not shown, however it isunderstood that maximum power handling capacity is obtained if both thebuses and the flexible cables are water-cooled. In FIG. 3a it is seenthat the bus tubes of each group are positioned in the form of atriangle and the centers of gravity 63, 64, and 65 of the A phase group27, the B phase group 28, and the C phase group 29 are shown to betriangularly positioned. Preferably, because of space limitations thisis an isosceles triangle in which the distance, measured along anysectional view parallel to line 33, between the centers 63, 64 is thesame as the distance between centers 64 and 65, and the distance betweencenters 63 and 65 is greater than either of the previously mentioneddistances between centers of gravity. The B phase hanger plate 43 ispositioned above the A and C phase hanger plates 36, 46, respectively, adistance corresponding to the height of the B phase transformer bus 18above the A and C phase transformer buses 17, 19, respectively. Withthis arrangement, a substantially triangular relationship exists betweenthe A, B and C phase conductor groups of flexible cables 23, 24 and 25which will be subsequently described in conjunction with FIG. 4. In FIG.3a, the spacing between bus tubes of a phase group is the same for eachof the phase groups. For example, the spacing between bus tubes 33, 34and 35 is identical to the spacing between bus tubes 39, 40 and 41 andthe spacing between bus tubes 48, 49 and 50, A more equal distributionof power to electrodes 20, 21 and 22 can be achieved by the bus tubeconfiguration shown in FIG. 3b. This is achieved by adjusting the selfgeometric mean distances within the phase group, inversely according tothe power distribution which obtains in a co-planar arrangement, whichself geometric mean distance may be otherwise termed the geometric meanradius of the phase group. For example, in the A phase tubes of thegroup 27, the geometric mean radius of the group is increased relativeto those of FIG. 3a by increasing the distance between conductors 33, 34and 35 as shown in FIG. 3b. The geometric mean radius of the B phasegroup conductors 28, is decreased as shown in FIG. 3b, in whichconductors 39, 40 and 41 were spaced more closely together than in thearrangement of FIG. 3a because the B phase or intermediate phase are, isthe hottest with a co-planar transmission system. The C phase groupconductors 29, namely conductors 48, 49 and 50, are maintained at thesame geometric mean distance as in FIG. 3a because the C phase electrodere- 9 ceives the intermediate amount of power in a co-planararrangement, as previously explained.

In other words, the geometric mean radius of a phase group of buses hasbeen modified in inverse relationship to the amount of power deliveredto the respective electrode by co-planar conductors. The electrode 21employing the system of buses of FIG. 3a would receive more power thanthe electrode 20. Similarly, with the system of 3a, the electrode 22would receive less power than the electrode 21 but more power than theelectrode 20. As shown in the geometric mean radius of the B phase groupof conductors 28 has been decreased by moving conductors 39, 40, and 41closer together. The resulting self inductance of the B phase group ofconductors has been increased. By moving the A phase group of conductorsfarther apart, or increasing the geometric mean radius of the A phasegroup of bus tubes, such that conductors 33, 34, and 35 are fartherapart than the C phase group of bus tubes 48, 49, and 50, the selfinductive reactance of the A phase group has been decreased. Thus, thebus arrangement indicated in FIG. 312 will result in a more equaldistribution of power to the electrodes 20, 21, and 22 than thearrangement of FIG. 3a. It is also understood that this substantiallyeven distribution of power is obtained at the three electrodes 20, 21and 22 because the total crosssectional area of the bus tubes and hencethe resistances are the same within each phase group and preferably thesame length of bus tubes is employed in the several phase groups 27, 28,and 29.

FIG. 4a is a view in section, taken along the lines 44 of FIG. 1 andshowing a preferred embodiment of cable arrangement of the flexiblecables in the A, B, and C phase groups 23, 24, and 25, respectively. Inthis particular arrangement, the geometric mean radii within the phasegroups have been adjusted in a manner similar to the adjustment of thebus tubes in FIG. 3b, namely inversely in accordance with the powerdelivered to the respective arcs by a co-planar arrangement of flexiblecables. For example, in a co-planar cable arrangement of phase groups,the flexible cables 52, 53, and 54 of the A phase group 23 normallycarry the least power to their electrodes 20. The phase groups are saidto be co-planar if the A phase group has its center of gravity 68coplanar With the center of gravity 69, 70 of the B and C phase groupsrespectively, as viewed in any normal section. Thus, the distancedesignated 0, in FIG. 4a, indicating the distances between each pair offlexible cables 52, 53, and 54 is greater than the distance indicated bythe letter b between the conductors 55, 56, and 57. Similarly, thedistances between the conductors 58, 59, and 60 indicated by the lettera are greater than the distances between the conductors 55, 56, and 57indicated by the letter b. In this embodiment, the geometric meandistances are changed by moving the intermediate phase group inproportion to the power delivered to the respective outer electrodeswith a co-planar arrangement. In this specification, the geometric meandistance is considered to be synonymous with the distance between thecenters of gravity of the phase groups. For example, the distanceindicated by the letter e between the centers of gravity 68, 69 of the Aand B phase groups is preferably less than the distance indicated by theletter 7 between the centers of gravity 69, 70 of the B and C phasegroups, respectively. The distance indicated by the letter d between thecenters of gravity 68, 70 of the A and C phase groups is larger thaneither of the distances designated e or f.

FIG. 4b is a view in section, taken along the lines 4-4 and showing analternative arrangement of flexible cables in the phase groups of cables23, 24, and 25. The A phase group includes four cables 75, 76, 77, and78 having a group center of gravity 79. The B phase group includes onlytwo cables, 82 and 83, having a center of gravity 84. The C phase groupincludes three cables 86, 87, and 88 having a center of gravity 89. Inthis particular example, the distances g and 12, between center ofgravity 34 and the centers of gravity 79, 89, respectively, are equaland greater than the distance i between centers of gravity 79, 89. Inother words, as compared with a co-planar arrangement of phase groups ofconductors, a positioning of the hottest arc phase group of conductorsequidistant from the A and C phase groups of conductors produces abetter distribution of power if the B phase GMR is reduced and the Aphase GMR is increased. This can be done by decreasing the number ofconductors in the B phase group and moving the remaining conductorscloser together and by increasing the number of conductors in the Aphase group. The GMR of the A phase group can be further increased bymoving the conductors 75, 76, 77, and 78 farther apart than the B phasegroup conductors 82, 83. Each of the cables is of the same length andcross-sectional area so that they are interchangeable, as in theprevious example shown in FIG. 4a.

Because of greater physical limitations on the bus section than on theflexible cable section, wider variations of GMRs and GMDs are possiblein the latter section to achieve substantially equal power distribution.Further, the GMR of a phase group of cables may be increased ordecreased by adding or removing interchangeable cables of the samelength and cross-section, as explained with respect to FIG. 4b. Thisprocedure, however, modifies resistance as well as rea-ctance within aphase group for the benefits and burdens thereof Within the precepts ofmy invention.

The above described principles have been successfully applied to atleast one electric furnace transmission system. In this illustrativeinstallation, power is supplied to three electrodes from atwelve-thousand kva. three-phase transformer through a transmissionsystem in which the outer leading phase group of conductors includesthree cables spaced with their axes at the centers of an equilateraltriangle fifteen inches on a side. The opposite outer phase group ofconductors includes a pair of cables vertically spaced fourteen inchesapart and with the center of gravity of this opposite outer phase groupspaced approximately forty-eight inches from the center of gravity ofthe first mentioned outer phase group. The middle phase group includes apair of cables positioned with their centers seven inches apart and withthe center of gravity of this group spaced two feet from the planecontaining the geometric centers of the outer phase groups ofconductors. These flexible cables are coupled to a system of buses, twofor each phase having a 2.875 outside diameter and a 1.771" insidediameter. The buses are positioned with their axes at the centers of atriangle, the base of which is thirty-one inches long at a pointadjacent the electrodes and the outer phase groups taper outwardly to adistance of fifty-four inches at the connection between the buses andthe flexible cables, and in which the lengths of the triangle betweenthe outer buses and the intermediate buses was approximately thirtyinches at the electrodes and increased to approximately forty-fiveinches at the connection of the buses to the flexible cables. The middlephase group of buses extend two feet beyond the plane containing theends of the outer phase buses to maintain the triangular relationship ofthe flexible cables throughout their length. The average geometric meandistances between the central phase group and the two outer phase groupswas approximately 37.5 inches and the geometric mean distance betweenthe outer phase groups of conductors was approximately 42 inches. Thetransmission system delivered power to the arcs which was computed to be4.58 megawatts to the phase A electrode, 4.68 megawatts to the B phaseelectrode and 4.74 megawatts to the C phase electrode. This distributionof power, and total, compares very favorably and shows a definiteimprovement over a co-planar arrangement of the same bus tubes and sameflexible cables in which the self geometric mean distances wereunchanged and in which the intermediate phase groups were positionedwith their centers of gravity midway between the outer phase groups.

The power delivered through this co-planar system to the electrodes wascomputed to be 4.39 megawatts, 4.83 megawatts, and 4.53 megawatts forthe A, B and C electrodes, respectively. Even a non-co-planararrangement of bus tubes with the co-planar arrangement of flexiblecables of the first described installation, resulted in an improvedcomputed distribution of power relative to the complete co-planarsystem. In this arrangement, the A, B and C phase electrode receivedpower computed to be 4.50, 4.74 and 4.64 megawatts, respectively.

I have determined that a flexible cable arrangement corresponding tothat shown in FIG. 4a may be employed in which the geometric mean radiiof the A, B and C phase groups of flexible conductors is 9.32 inches,8.15 inches and 6.84 inches, respectively. The geometric mean distancesbetween the A and B phase groups is 48 inches, the geometric meandistance between the B and C phase groups is 48 inches and the geometricmean distance between the A and C phase groups is 60 inches. The bustube arrangement corresponds to that shown in FIG. 2 and may employ anaverage geometric mean distance between the A and C phase groups of 69inches, an average geometric mean distance between the B phase group andthe A and C phase groups of 53 inches. The individual tubes 33, 34 andof the A phase group comprised three six-inch copper tubes spaced withtheir centers ten inches apart at the corners of a trianglesubstantially throughout their length such that the self geometric meandistance or geometric mean radius, for each of the phase groups, is 6.82inches. Employing a transformer rated at 65,000 kva. this transmissionsystem produces a computed power distribution of 14.95 megawatts, 16.46megawatts, and 16.07 megawatts to the A, B and C phase electrodes,respectively.

While I have illustrated and described a preferred and other form of myinvention, modifications, changes and improvements therein and thereonwill occur to those skilled in the art who come to practice my inventionand understand the precepts and teachings of this specification.Therefore, I do not want to be restricted in the scope and effect of mypatent to any of the forms herein specifically disclosed, nor in anyother manner inconsistent with the progress by which the art has beenpromoted by my invention.

What is claimed is:

1. An electric power transmission system for an electric arc furnaceincluding a poly-phase transformer having a primary winding, a secondarywinding and an output terminal for each phase of the poly-phase system;

a furnace electrode for each phase; and non-coplanar phase groups ofconductor means connecting each of each output terminals to one of saidelectrodes, each of said conductor means having substantially the samecross-sectional area, one of said phase groups having a differentgeometric mean radius from another of said phase groups.

2. The system of claim 1 in which each of said conductor means comprisesa flexible portion of substantially the same length as the flexibleportion of every other conductor means.

3. The system of claim 2. in which the geometric mean distance betweenone pair of said phase groups is different fro-m the geometric meandistance between another pair of said phase groups.

4. The system of claim 3 in which each of said conductor means alsocomprises rigid bus means connecting said flexible portions to saidelectrode.

5. The system of claim 4 in which the geometric means distance betweenpairs of groups of said rigid bus means is at least in parts of theirlengths different from the geometric mean distances betweencorresponding pairs of said flexible portions.

6. The system of claim 1 in which the said phase groups are respectivelyleading, middle and lagging, and, for the transmission of substantiallyequal power to the electrodes, the said geometric mean distance betweenthe leading and middle pair of groups is smallest, and the geometricmean distance between the leading and lagging pair is largest.

7. The system of claim 6 in which the geometric mean radius for the saidleading phase is greater than the geometric mean radius of each of theother phase groups and the geometric mean radius of the said middlephase group is less than that of the other phase groups.

8. The system of claim '7 in which each of said conductor meanscomprises a flexible portion substantially equal in length to theflexible portions of other conductor means.

9. The transmission system according to claim 1 wherein each group ofconductor means has a different geometric mean radius.

10. The system of claim 1 in which said phase groups are triangularlyrelated and the geometric mean distances between each pair of phasegroups are different.

11. The system of claim It) in which the said phase groups arerespectively leading, middle and lagging, and, for the transmission ofsubstantially equal power to the electrodes, the said geometric meandistance between the leading and middle pair of groups is smallest, andthe geometric mean distance between the leading and lagging pair islargest.

12. The system according to claim 11 in which the geometric mean radiusfor the said leading phase is greater than the geometric mean radius ofeach of the other phase groups and the geometric mean radius of the saidmiddle phase group is less than that of the other phase groups.

13. The system according to claim 12 in which said conductor means alsocomprises rigid bus portions connecting said flexible portions to saidelectrodes, said bus portions being arranged in the same phase groups assaid flexible portions and having at least in part of their lengthssubstantially the same geometric mean distances between pairs of groupsas said flexible portions.

14. The system of claim 9 in which the said phase groups arerespectively leading, middle and lagging, and, for the transmission ofsubstantially equal power to the electrodes, the geometric mean distancebetween the leading and middle pair of groups is smallest, and thegeometric mean distance between the leading and lagging pair is largest.

15. The system according to claim 14 in which the geometric mean radiusfor the said leading phase is greater than the geometric mean radius foreach of the other phase groups and the geometric mean radius of the saidmiddle phase group is less than that of the other phase groups.

16. The system of claim 15 in which the longitudinal centers of saidphase groups are arranged approximately in the respective apexes of atriangle having sides of three different lengths, and each of saidconductor means comprises a flexible portion substantially equal to theflexible portions in the other conductor means, and at least one of saidphase groups has a different number of said flexible portions thananother of said phase groups.

17. In a high-current three phase AC transmission system fortransmitting power from the three output terminals of a transformer tothe corresponding electrodes of an electric furnace wherein a phasegroup of conductor means is connected between each said terminal andeach electrode, and each group comprises flexible portions; theimprovement comprising that the said flexible portions of each phasegroup are disposed in triangular relation one to the other and that forsubstantially equal power transmission to each electrode the geometricmean distance between the pair of groups which tend to transmit the mostand least power, respectively, is less than the geometric mean distancebetween the pair of groups which tend to transmit the greatest andmiddle power respectively. 18. The improvement of claim 17 in which saidconductor means also comprises rigid bus means having at least portionsrelated to each other differently than said flexible portions andtending to unbalance the distribution of power transmitted in thediflFerent phases.

19. The improvement in claim 18 wherein the geometric mean radius of thesaid flexible portions of the phase group tending to transmit the mostpower is less than the geometric mean radius of the flexible portion ofphase group tending to transmit the least power.

20. The improvement according to claim 19 wherein the said flexibleportions comprise equal cables of equal length and equal impedence andthe number of said cables in the said group which tends to transmit themost power is less than the number of said cables in the group whichtends to transmit the least power.

21. The improvement of claim 17 in which the said first and second namedgeometric mean distances comprise first and second sides of the saidtriangular relation and the third side corresponds to the geometric meandistance between the pair of groups which tends to transmit the leastand middle power respectively, and said improvement also comprises thateach said geometric mean distance is changeable without altering eitherof the others.

22. The improvement of claim 17 wherein said flexible portions comprisecables of substantially equal length and equal impedance.

23. The improvement of claim 22 wherein the number of cables in thephase group tending to transmit the most power is less than the numberof cables in another phase group which tends to transmit less power.

24. The improvement of claim 23 wherein the geometric mean radius of thephase group tending to transmit the most power is less than thegeometric mean radius of another group which tends to transmit lesspower.

25. The method of transmitting substantially equal power by each ofthree phase groups of conductors in an AC system from a transformer toeach of three electrodes of an electric furnace comprising,

5 arranging substantial portions of the lengths of each of said groupsat the three corners respectively of a triangle wherein the three pairsof said groups define the sides of said triangle, and

reducing the geometric mean distance between the pair of groups whichtend to transmit the most and least power respectively withoutproportionately altering the geometric mean distance between the groupswhich tend to transmit the most and median power respectively.

26. The method of claim 25 with the additional step of changing therelative geometric mean radii of said first named pair of groups.

27. The method of balancing power according to claim 26 wherein the stepof adjusting the geometric mean 20 radius includes the step of changingthe number of conductors in one of said phase groups.

References Cited UNITED STATES PATENTS BERNARD A. GILHEANY, PrimaryExaminer.

R. N. ENVAL, JR., Assistant Examiner.

1. AN ELECTRIC POWER TRANSMISSION SYSTEM FOR AN ELECTRIC ARC FURNACEINCLUDING A POLY-PHASE TRANSFORMER HAVING A PRIMARY WINDING, A SECONDARYWINDING AND AN OUTPUT TERMINAL FOR EACH PHASE OF THE POLY-PHASE SYSTEM;A FURNACE ELECTRODE FOR EACH PHASE; AND NON-COPLANAR PHASE GROUPS OFCONDUCTOR MEANS CONNECTING EACH OF EACH OUTPUT TERMINALS TO ONE OF SAIDELECTRODES, EACH OF SAID CONDUCTOR MEANS HAVING SUBSTANTIALLY THE SAMECROSS-SECTIONAL AREA, ONE OF SAID PHASE GROUPS HAVING A DIFFERENTGEOMETRIC MEAN RADIUS FROM ANOTHER OF SAID PHASE GROUPS.